Global Electron Thermodynamics in Radiatively Inefficient Accretion Flows

The global dynamics of the ions and electrons in an accretion flow where ion-electron collisions are negligible can be described by conservation of mass and conservation of energy independently in each species (see, e.g., Ressler et al., 2015). The mass conservation is given by

We consider a charge-neutral hydrogen plasma such that number densities of electrons and ions are equal, $n_{e}\approx n_{i}$, and are related to the mass densities by $\rho_{e}=m_{e}n_{e}$ and $\rho_{i}=m_{i}n_{i}$, where $m_{e}$ and $m_{i}$ are the electron and ion masses, respectively. We further assume that both the ions and electrons follow the same flow four-velocity, i.e., $u^{\mu}_{i}\approx u^{\mu}_{e}\equiv u^{\mu}$ and that all flow quantities are azimuthally symmetric.

Hereafter, we will express the mass accretion rate in terms of the Eddington critical rate

where $\sigma_{T}$ is the Thomson cross section and we set the radiative efficiency of the accretion flow $\eta$ to be 0.1. We also set the scale height by imposing local vertical pressure balance at each radius (Özel et al., 2022), i.e.,

where $u^{\phi}$ is the azimuthal angular velocity of the flow, assumed to be equal to the local Keplerian value.

Using the form of the stress energy tensor defined in Equation (4), we simplify the conservation of energy in Equation (3) to obtain

Because of the azimuthal and vertical averaging of the equations, as well as the implicit averaging over the coherence time of the MHD turbulence, the various components of the fluid velocity describe only the overall motions of the plasma and not its fluctuating turbulent motions. Due to the same averaging, both the pressures, $P_{i}$ and $P_{e}$, and the effective internal energies, $\epsilon^{\rm{eff}}_{i}$ and $\epsilon^{\rm{eff}}_{e}$, include contributions from the fluctuating turbulent plasma motions. Note that the magnetic field does not enter explicitly the energy equations.

Following the approach in Özel et al. (2022), we write the radial velocity outside of the innermost stable circular orbit (ISCO) as

where $\eta$ and $n_{R}$ are parameters that aim to capture the efficiency of the mechanism that drives angular momentum transport in the flow. By exploring a range for these parameters, we allow our model to be general enough to encompass different plausible dissipation profiles.

Using this model for the radial velocity, we can easily apply mass conservation (equation [5]) and obtain a radial profile for density, which yields another power law function in radial distance.

In order to infer the temperatures of each species, we employ an ideal gas equation of state for the ion and electron thermal pressures. Adding the contributions of the fluctuating turbulent motions, to which we assign an r.m.s. velocity of $u_{\rm turb}$, we write the effective pressure of each species as

Under the reasonable assumption that the r.m.s. turbulent velocities scale with the local Alfvén velocity, $u_{\rm turb}^{2}=\zeta u_{\rm A}^{2}=\zeta B^{2}/(4\pi m_{\rm i}n)$, where $\zeta$ is a factor of order unity, the equation for the effective pressure of the electrons becomes

In contrast, the equation for the effective pressure of the ions becomes

Similarly, the effective internal energy becomes

for the electrons and

for the ions. In all these equations, we have defined the plasma parameter as $\beta=8\pi P_{i}/B^{2}$.

Allowing for trans-relativistic behavior of the ions and electrons, we express the temperature dependence of the adiabatic index of a gas as (see Hakim, 2011)

where $H=K_{3}(1/\Theta)/K_{2}(1/\Theta)$. In this equation, $K_{n}$ denotes the modified Bessel function of the second kind at $n^{\rm{th}}$ order and the dimensionless temperature for the two species is given by $\Theta_{(s)}\equiv k_{B}T_{(s)}/m_{(s)}c^{2}$.

The viscous stresses present in the accretion flow channel the gravitational potential energy into turbulent wave energy that resides as MHD waves in the disk. These MHD waves, in principle, undergo dissipation at microscopic length-scales and channel their energy into the thermal energies of the plasma species (ions and electrons). Hence, the heating rates of the individual species, $Q_{i}$ and $Q_{e}$, are determined by two factors: (i) the total energy dissipated in the plasma due to viscous stresses (defined as $Q\equiv Q_{i}+Q_{e}$) and (ii) the partition of the total dissipated energy into the ions and electrons (defined as $f\equiv Q_{i}/Q_{e}$).

In addition to direct heating, there exists a channel of energy exchange between the ions and the electrons through Coulomb collisions ($Q_{ie}$, see equations [8] and [9]), which is given by (see Colpi et al., 1984)

In a radiatively inefficient flow, the electrons cool primarily via synchrotron and Bremsstrahlung radiation (see Yuan & Narayan, 2014). The synchrotron emissivity ($j_{\nu}$) for temperatures above the trans-relativistic regime can be written as (Mahadevan et al., 1996)

When the optical depth is low, the electron cooling rate, $\Lambda_{e}$, is simply the emissivity integrated over all frequencies and is given by

We note that, while the rate of viscous heating is directly proportional to the plasma density, the resultant energy transfer rates due to Coulomb collisions, Bremsstrahlung cooling, and synchrotron cooling are proportional to the square of the plasma density, introducing an overall scale which is linear in plasma density into the equations (8) and (9). This is why the overall mass accretion rate does not drop out of Equations (8) and (9) and needs to be specified, unlike the case in adiabatic GRMHD simulations.

We first solve for the ion and electron temperatures in a flow with a mass accretion rate $\dot{M}\ll 10^{-7}\dot{M}_{E}$. In this limit of extremely low mass accretion rate, turbulent heating dominates over the effects of Coulomb heating, while radiative cooling is negligible. We also set the strength of the magnetic field in the plasma by choosing the fiducial value $\beta=5$, the radial velocity profile by setting its power-law index to $n_{R}=1.5$, and the value of $\zeta=0.2$ for all our calculations.

Figure 2 shows the radial profiles of the ion and electron temperatures, as well as their ratio. As they drift inwards, both the ions and the electrons quickly heat up from their initial sub-virial temperatures at the outer radial boundary to the local virial temperatures. The ions then maintain a virial, $r^{-1}$, profile all the way to the inner region of the accretion flow. On the other hand, the electron temperature gradually shifts from one virial profile to another with a lower normalization in the inner accretion flow.

In order to understand these solutions, it is helpful to examine an analytical limit of the solutions to equations (8) and (9), where the effects of Coulomb heating and radiative cooling are neglected and the turbulent heating rate is approximated only to leading order, assuming that it is independent of the temperature ratio $T_{i}/T_{e}$. We further simplify the equations by eliminating the temperature dependence of the effective adiabatic indices $\gamma_{i}$ and $\gamma_{e}$. In this limit, following the Appendix of Özel et al. (2022), we write the ion temperature as

and the electron temperature as

Having established the scale-free virial nature of ion and electron temperatures under turbulent heating, we now examine the effects of Coulomb coupling between ions and electrons on the flow thermodynamics. Figure 3 shows the temperatures at density scales corresponding to mass accretion rates of $\dot{M}=10^{-7}\dot{M}_{E}\ \rm{and}\ 10^{-4}\dot{M}_{E}$ and also compares with the solution previously calculated with $\dot{M}\ll 10^{-7}\dot{M}_{E}$.

At $\dot{M}=10^{-7}\dot{M}_{E}$, Coulomb collisions cause the ion and electron temperatures to equilibrate at large radii. As the collision cross section decreases with increasing temperature, the two species decouple inwards and approach their individual virial solutions set by turbulent heating. Upon further increasing the accretion rate to $\dot{M}=10^{-4}\dot{M}_{E}$, and consequently the density scale, Coulomb coupling becomes stronger. As a result, the ions and electrons remain in thermal equilibrium at even smaller radii. However, even at such accretion rates, the effects of Coulomb coupling fade away as the gas drifts inwards, with the ion and electron temperatures approaching the limit of their scale-free solutions determined by turbulent heating in the inner flow.

The above analysis of the relative importance of turbulent and Coulomb heating indicates that, at density scales relevant for radiatively inefficient accretion flows, turbulent heating almost completely overwhelms the effects of Coulomb interactions in the inner regions of the accretion flow. While we will examine the effects of radiative cooling in a later subsection, it is instructive to consider certain implications of the scale-free turbulent heating of electrons.

The initial models of radiatively inefficient accretion flows (see Narayan & Yi, 1995b) considered the electrons to be heated solely via Coulomb coupling with the ions, without any of the turbulent energy in the plasma being channeled directly into the electrons. We compare such a system with our set-up of direct electron heating in Figure 4, where we show the ion and electron temperatures for a system where the electron heating is purely through Coulomb collisions with the ions. We also compare it with the direct heating model described in this paper, i.e., where a fraction of the turbulent energy is directly channeled into the electrons. The ion temperatures expectedly follow virial profiles for both these cases since the channel for ion heating remains the same. However, the electron temperatures are no longer virial, when electron heating is purely through Coulomb collisions. Instead, when the temperature increases to the point that Coulomb interactions become negligible, electrons experience pure compressional heating ($T_{e}\sim n_{e}^{\gamma-1}$, shown in the dotted line in Figure 4), which results in a significantly shallower radial profile. The consequence is an ion-to-electron temperature ratio that increases rapidly with decreasing radius to values $\sim 100$. This comparison demonstrates that, when microphysical plasma phenomena channel a finite amount of the turbulent wave energy into the electrons, this introduces substantial changes to the nature of the solution for the electron temperature compared to a setting where electrons only receive energy via Coulomb heating.

Having established the relative importance of particle heating via turbulent heating and Coulomb collisions, we now turn to examining the effects of electron cooling on the thermodynamics of the plasma. As described in §2.4, we include the effects of Bremsstrahlung and synchrotron cooling of the electrons in evolving the electron thermodynamics in the accretion disk. The plasma cooling rates due to both ion-electron and electron-electron Bremsstrahlung is proportional the square of the number density of the plasma ($\Lambda_{\rm{b}}\propto n_{e}^{2}$). On the other hand, the synchrotron cooling rate scale as $\Lambda_{e}\sim n_{e}B^{2}$ in our regime of interest. Because we set the local magnetic field to be proportional to the local thermal pressure using the plasma $\beta$ parametrization, the synchrotron cooling rate effectively scales with $n_{e}^{2}$, which is the same as Bremsstrahlung cooling and Coulomb heating (see appendix of Satapathy et al., 2022).

Figure  5 shows the effects of radiative cooling processes at different accretion rates on the electron temperature and the ion-to-electron temperature ratio in the inner accretion disk. At an accretion rate of $10^{-7}\dot{M}_{E}$, radiative cooling is inefficient and the electron temperatures are equal to the case with $\dot{M}\ll 10^{-7}\dot{M}_{E}$, for which radiative processes have been neglected. Only at higher accretion rates, such as $\dot{M}=10^{-4}\dot{M}_{E}$, there is a small drop in the electron temperature. It is important to emphasize here, however, that because of the particular dependence of the cooling rates on the square of the density, the electron temperatures retain their virial profiles.

The effects of synchrotron cooling also depend significantly on plasma $\beta$, which sets the strength of the magnetic field. However, since the partition of turbulent heating between ions and electrons also depends strongly on plasma $\beta$, we will address the combined effect in the following subsection.

In the previous subsections, we established the near-virial nature of the ion and electron temperature profiles in the inner accretion flow and addressed their weak dependence on the efficiency of Coulomb collisions and radiative cooling processes. Because the ions and the electrons receive their heat mostly from turbulent dissipation in the accretion flow, the microphysical processes that partition the turbulent energy into the individual species are extremely important in determining their temperature ratios. As discussed in §2.3.2 and shown in Figure 1, the partition of heat in the regime of plasma $\beta\leq 10$ depends strongly on the composition of the turbulent cascade, which is not well understood for MRI-driven turbulence.

In this subsection, we study the dependence of the electron temperatures on the composition of the turbulent cascades, for the regime of interest in plasma $\beta$. In particular, we will focus on the asymptotic temperature ratio $T_{e}/(T_{e}+T_{i})$ in the innermost regions of the accretion flow, which measures the relative contribution of the electrons to the total gas pressure. Figure 6, shows this ratio calculated at $r=10\;GM/c^{2}$, as a function of plasma $\beta$, for a range of mass accretion rates of $10^{-7}\leq\dot{M}/\dot{M}_{E}\leq 10^{-5}$, and for different ratios of the compressive-to-Alfvenic wave power.

The widths of the shaded regions in Figure 6 represent the dependence of the temperature ratio on the accretion rate, which is determined primarily by the efficiency of synchrotron cooling. At low plasma $\beta$, where the magnetic field is stronger, synchrotron cooling is very efficient and introduces a factor of $\sim 2$ range of temperature ratios for the accretion rates we consider here, as discussed in §2.4. In the opposite limit of high plasma $\beta$, synchrotron cooling becomes inefficient and the temperature ratio in the inner accretion flow becomes practically independent of accretion rate.

The dependence of the temperature ratios on plasma $\beta$ simply reflects the dependence of heat partition between ions and electrons in the turbulent cascade shown in Figure 1. The ratio of compressive-to-Alfvenic wave power affects primarily the temperature ratio in the low-$\beta$ regime, for which the ion-to-electron heating ratio asymptotes to $P_{\rm comp}/P_{\rm A}$, for reasons discussed earlier. In order for the turbulent cascades to heat predominantly the electrons and not the ions, i.e., leading to $T_{i}<T_{e}$, the MRI-driven turbulence would need to be purely Alfvenic.

For the latter case of purely Alfvenic turbulence, Figure 6 also compares the temperature ratios obtained using the results of the numerical model of Alfvenic cascades (K2020 with $P_{\rm{comp}}/P_{A}=0$) to those using the analytic calculation of Howes (2010). The nonlinear effects captured in the numerical calculation of the cascades lead to smaller electron temperatures, in the intermediate plasma $\beta$ range but to larger electron temperatures in the very high plasma $\beta$ regime compared to the earlier studies.

In earlier calculations of the observational properties of radiatively inefficient flows based on GRMHD simulations, the ratio of the ion-to-electron temperatures have often been prescribed in post-processing, using different empirical relations that depend on plasma $\beta$ (see, e.g., Chan et al., 2015; Mościbrodzka et al., 2016). In particular, the empirical relation

has been used in comparing GRMHD model predictions to EHT observations (see Akiyama et al., 2022b). Here $R_{\rm{high}}$ is the temperature ratio in the limit of high plasma $\beta\gg 1$, and $R_{\rm{low}}$ is the temperature ratio in the limit of low plasma $\beta\ll 1$.

Figure 6 compares this empirical relation to our calculation of the temperature ratio. For the empirical ratio, we have set $R_{\rm{low}}=1$, as has been typically adopted. We find that, at the high plasma $\beta$ limit, the asymptotic temperature ratio corresponds to $R_{\rm high}\simeq 40$. However, the functional dependence of the temperature ratio on plasma $\beta$ is not captured by empirical relation that is typically employed. Moreover, the low plasma $\beta$ limit given by $R_{\rm low}$ does not capture the dependence of the temperature ratio on the composition of the turbulent cascade.

Our model of the accretion flow allows us to generate a new, physically motivated empirical relation for the ion-to-electron temperature ratio. Taking the ratio of equations (34) and (35), we obtain

Suppressing the weak dependence on temperature of the partition of heat between ion and electrons in equation (26), we write

We estimate the effective adiabatic index of the ions using equation (15) to write

Because the ions remain subrelativistic throughout the flow, we set $\gamma_{i}=5/3$.

As discussed above, the temperature of the electrons and, hence, their adiabatic index, depends on the value of plasma $\beta$. At low values of plasma $\beta$, the electrons get to sufficiently high tempeartures to become ultra relativistic. In this limit, their adiabatic index is $\lim_{\beta\rightarrow 0}\gamma_{e}=4/3$. In the regime of high plasma $\beta$, the electron temperature is significantly lower and their adiabatic index increases towards the non-relativistic value of 5/3. In principle, one could use expression (17) for the temperature dependence of the adiabatic index and solve the implicit equation for the electron temperature. For computational efficiency, we have devised an empirical relation for the adiabatic index of the electrons in the inner accretion flow using the form

We have then set $\kappa=0.13$ and $\beta_{\rm{break}}=5$ in order to reproduced the behavior seen in the detailed solutions of our accretion model.

Figure 7 compares the ratio of the electron-to-total gas temperatures calculated using our covariant semi-analytical model of the accretion flow to the empirical relation (37), using expressions (38)–(40) for its parameters. For this comparison, we have set the velocity profile parameter to $n_{R}=1.5$ and neglected the effects of radiative cooling. The latter are expected to be relatively unimportant for the accretion flows around the two prime imaging targets of the EHT, Sgr A* and M87.

The empirical relation (37) can be easily implemented in a calculation of observables based on postprocessing of GRMHD simulations. The $n_{R}$ velocity profile parameter can be inferred empirically from the time- and azimuthally-averaged velocity (or density) profile in the simulation. Finally, the ratio of the compressive-to-Alfvenic wave power can be either set to the value inferred by Kawazura et al. (2022) or allowed to be a free parameter to be inferred observationally.